I personally start looking at the speed of the breath and then it slows down. In the Tripitaka are there recommended methods to accomplish this? What do different teachers and commentaries say about it?

One meditation teacher I found who emphasizes calming of the bodily is Bhante Vimalaramsi in 6Rs. But exact mechanics is not really covered on how the relaxation happens. Does this also lead to or linked to calming of the bodily fabrications? Are there more explicit instructions, especially those we can source from the Tripitaka?

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The bodily fabrication, i.e. the in & out breath, is said to calm considerably as one progresses through the first 3 steps mentioned in the Anapanasati Sutta, and it does.

The first 2 steps, requires an evaluation of the breath (not control it), to help the mind be mindful of the breath, and take it as it meditation object.

Aware that the in-breath is long one breathes in / aware that the out-breath is long one breathes out

Aware that the in-breath is short one breathes in / aware that the out-breath is short one breathes out

As mindfulness of the breath is established, the practitioner will be able to experience his/her whole body (sabba kaaya), albeit subtly, as the inhaling and exhaling motions generate sensations throughout the body. This will take some time to develop, before one experiences the entire body gently pulsating as the breath comes in and goes out. The upper body is usually more prominent at first. Eventually one should be able to feel sensations in the lower body. It could feel like floating on a lake with a gentle ripple.

Those sitting on chairs, may take longer to experience the lower body due to the posture, than someone sitting cross legged on a cushion of floor.

As one continues to be mindful of these subtle whole-body sensations, one will begin to be aware that the breath (kaaya sankhara, i.e. the bodily formation) is now calmer, more subtle, than before, sometimes its completely still for a while.

So, I would think that the recommended method in the Sutta can be found in the Anapanasati sutta mainly.

In addition, the sutta's mention about the importance of posture and preparation

A place conducive for meditation and free from disturbances (wilderness, to the shade of a tree, or to an empty building)

The cross-legged position

Keeping the body erect

Non remorse due to having done immoral acts in the past. So keeping one's virtue and being moral is extremely useful.

Also the starving of the panca nivarana (five mental hindrences). Bhikkhu Nyanaponika's book contains direct references from the suttas on how to do so.

The Pali should be translated as "body fabricator". MN 44 defines the body fabricator as the breathing in & out. Calming the kaya sankhara means calming the breathing. That simple. The breathing is calmed by observing it; that is; by abandoning craving so the mind become clear & naturally observant.

by "playing", using, testing out kinds of breath and perceptions, feelings, touches around it, one clams the body fabrications together with the related mind fabrications till joy within the body, and related calming of domanassa is experienced. Dwelling right there one, dwells in the first Jhana.

But such can not be done when still attached to home, the world, to the senses or when there is much remorse by not having taken the precepts by heart. If one is still a stingy person, or one without gratitude, wrong views, such as well is a total blocking hindrance. So a lot of merits in advanced are required and will give the causes to do such meditation.

The fourth step is, in the Buddha’s words, to calm bodily fabrication. You tell yourself, every time you breathe in, “I will breathe in calming bodily fabrication.” When you breathe out, you tell yourself, “I will breathe out calming bodily fabrication.” The word “fabrication,” or saṅkhāra, here has to do with what you are intentionally doing in the present moment to shape your experience. The term “bodily fabrication” means the breath, because there is an element of intention in the way you breathe, and this element of intention has an effect on the way you experience your body as a whole. As you begin to see the effect that the breath has on the whole body, you take advantage of the fact that you have the choice to make that effect more calming throughout the body.

Now, as he says elsewhere, one good way to calm the body is first to develop a sense of rapture—an energizing sense of fullness in the body. Then, when the body has been nourished by rapture, it will calm down, not from being forced to be calm, but because it feels naturally inclined to grow calm from that sense of fullness. So “calming bodily fabrication” can sometimes mean that you use the way you breathe to develop a sense of rapture in the body before you try calming it down. The sense of fullness and calm will then help you to stay more firmly in the present.

That's by the way also a tool how to prevent oneself in daily live, being touched by thing causing annoying or greed. By focusing on the breath for some deep breath in breath out one prevents oneself to act again foolish having uprooted the causes for foolish reactions within oneself.

This is why wise say "as soon as forgetting the breath one is like an orphan and acts in very foolish ways". Take care of you by not staying a poor orphan!

(Note that this is not given for trade, exchange, stacks, entertainment and akusala deeds, but as a share of merits and to continue such for release)

Bodily, verbal and mind fabrications can be calmed through breath watching. Humans have three centers - body, mind and heart. These centers and breath are interlinked. If your mind is at peace breathing will slow and deeper and if your mind is restless breathing will be shallow and fast. Buddha's terminology is slightly different. Verbal fabrications have their origin in mind and mind fabrications origin lies in Heart. All three fabrications can be calmed through Vipassna. But one needs to follow percepts in spirit. Trouble is lot of people are trying out meditation technique without becoming a true disciple of Buddha. Meditation is about understanding and implmenting messages of enlightened souls such as Buddha.